Signal Processing First Lab 02: Introduction to Complex Exponentials Multipath. x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt }

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1 Signal Processing First Lab 02: Introduction to Complex Exponentials Multipath Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises in the Pre-Lab section before going to your assigned lab session. Verification: The Warm-up section of each lab must be completed during your assigned Lab time and the steps marked Instructor Verification must also be signed off during the lab time. One of the laboratory instructors must verify the appropriate steps by signing on the Instructor Verification line. When you have completed a step that requires verification, simply demonstrate the step to the TA or instructor. Turn in the completed verification sheet to your TA when you leave the lab. Lab Report: It is only necessary to turn in a report on Sections 5 and 6 with graphs and explanations. You are asked to label the axes of your plots and include a title for every plot. In order to keep track of plots, include your plot inlined within your report. If you are unsure about what is expected, ask the TA who will grade your report. 1 Introduction The goal of this laboratory is to gain familiarity with complex numbers and their use in representing sinusoidal signals such as x(t) = A cos(ωt + φ) as complex exponentials z(t) = Ae jφ e jωt. The key is to use the appropriate complex amplitude together with the real part operator as follows: x(t) = A cos(ωt + φ) = Re{Ae jφ e jωt } 2 Overview Manipulating sinusoidal functions using complex exponentials turns trigonometric problems into simple arithmetic and algebra. In this lab, we first review the complex exponential signal and the phasor addition property needed for adding cosine waves. Then we will use MATLAB to make plots of phasor diagrams that show the vector addition needed when adding sinusoids. 2.1 Complex Numbers in MATLAB MATLAB can be used to compute complex-valued formulas and also to display the results as vector or phasor diagrams. For this purpose several new MATLAB functions have been written and are available on the SP First CD-ROM. Look for the MATLAB Files link just below the link for this lab. Make sure that this toolbox has been installed 1 by doing help on the new M-files: zvect, zcat, ucplot, zcoords, and zprint. Each of these functions can plot (or print) several complex numbers at once, when the input is formed into a vector of complex numbers. For example, the following function call will plot five vectors all on one graph: zvect( [ 1+j, j, 3-4*j, exp(j*pi), exp(2j*pi/3) ] ) 1 Correct installation means that the spfirst directory will be on the MATLAB path. Try help path if you need more information. CD-ROM SP First MATLAB Toolbox 1

2 Here are some of MATLAB s built-in complex number operators: conj abs angle real imag i,j x = 3 + 4i exp(j*theta) Complex conjugate Magnitude Angle (or phase) in radians Real part Imaginary part pre-defined as 1 i suffix defines imaginary constant (same for j suffix) Function for the complex exponential e jθ Each of these functions takes a vector (or matrix) as its input argument and operates on each element of the vector. Notice that the function names mag() and phase() do not exist in MATLAB. 2 When unsure about a command, use help. 2.2 Sinusoid Addition Using Complex Exponentials Recall that sinusoids may be expressed as the real part of a complex exponential: { x(t) = A cos (2πf 0 t + φ) = Re Ae jφ e j2πf 0t } (1) The Phasor Addition Rule shows how to add several sinusoids: x(t) = N A k cos(2πf 0 t + φ k ) (2) assuming that each sinusoid in the sum has the same frequency, f 0. This sum is difficult to simplify using trigonometric identities, but it reduces to an algebraic sum of complex numbers when solved using complex exponentials. If we represent each sinusoid with its complex amplitude Then the complex amplitude of the sum is X s = X k = A k e jφ k (3) N X k = A s e jφ s (4) Based on this complex number manipulation, the Phasor Addition Rule implies that the amplitude and phase of x(t) in equation (2) are A s and φ s, so x(t) = A s cos(2πf 0 t + φ s ) (5) We see that the sum signal x(t) in (2) and (5) is a single sinusoid that still has the same frequency, f 0, and it is periodic with period T 0 = 1/f 0. 2 In the latest release of MATLAB a function called phase() is defined in a rarely used toolbox; it does more or less the same thing as angle() but also attempts to add multiples of 2π when processing a vector. 2

3 2.3 Harmonic Sinusoids There is an important extension where x(t) is the sum of N cosine waves whose frequencies (f k ) are different. If we concentrate on the case where the (f k ) are all multiples of one basic frequency f 0, i.e., f k = kf 0 (HARMONIC FREQUENCIES) then the sum of N cosine waves given by (2) becomes x h (t) = { N N A k cos(2πkf 0 t + φ k ) = Re } X k e j2πkf 0t (6) This particular signal x h (t) has the property that it is also periodic with period T 0 = 1/f 0, because each of the cosines in the sum repeats with period T 0. The frequency f 0 is called the fundamental frequency, and T 0 is called the fundamental period. (Unlike the single frequency case, there is no phasor addition theorem here to combine the harmonic sinusoids.) 3 Pre-Lab Do all the exercises in this section before attending the regularly scheduled lab section. 3.1 Complex Numbers This section will test your understanding of complex numbers. Use z 1 = 10e j2π/3 and z 2 = 5 + j5 for all parts of this section. (a) Enter the complex numbers z 1 and z 2 in MATLAB. Plot them with zvect() and print them with zprint(). When unsure about a command, use help. Whenever you make a plot with zvect() or zcat(), it is helpful to provide axes for reference. An x-y axis and the unit circle can be superimposed on your zvect() plot by doing the following: hold on, zcoords, ucplot, hold off (b) The function zcat() can be used to plot vectors in a head-to-tail format. Execute the statement zcat([j,-1,-2j,1]); to see how zcat() works when its input is a vector of complex numbers. (c) Compute z 1 + z 2 and plot the sum using zvect(). Then use zcat() to plot z 1 and z 2 as 2 vectors head-to-tail, thus illustrating the vector sum. Use hold on to put all 3 vectors on the same plot. If you want to see the numerical value of the sum, use zprint() to display it. (d) Compute z 1 z 2 and plot the answer using zvect() to show how the angles of z 1 and z 2 determine the angle of the product. Use zprint() to display the result numerically. (e) Compute z 2 /z 1 and plot the answers using zvect() to show how the angles of z 1 and z 2 determine the angle of the quotient. Use zprint() to display the result numerically. (f) Compute the conjugate z for both z 1 and z 2 and plot the results. In MATLAB, see help conj. Display the results numerically with zprint. 3

4 (g) Compute the inverse 1/z for both z 1 and z 2 and plot the results. Display the results numerically with zprint. (h) Make a 2 2 subplot that displays the following four plots in one figure window: (i) z 1 and z 2 (ii) z 1 and z 2 on the same plot; (iii) 1/z 1 and 1/z 2 on the same plot; and (iv) z 1 z 2. Add a unit circle and x-y axis to each plot for reference. 3.2 ZDrill There is a complex numbers drill program called: zdrill CD-ROM Appendix A Demos ZDrill which uses a GUI to generate complex number problems and check your answers. Go to Appendix A demos to download and run. Please spend some time with this drill since it is very useful in helping you to get a feel for complex arithmetic. 3.3 Vectorization The power of MATLAB comes from its matrix-vector syntax. In most cases, loops can be replaced with vector operations because functions such as exp() and cos() are defined for vector inputs, e.g., cos(vv) = [cos(vv(1)), cos(vv(2)), cos(vv(3)),... cos(vv(n))] where vv is an N-element row vector. Vectorization can be used to simplify your code. If you have the following code that plots a certain signal, M = 200; for :M x(k) = k; y(k) = cos( 0.001*pi*x(k)*x(k) ); end plot( x, y, ro- ) then you can replace the for loop and get the same result with 3 lines of code: M = 200; y = cos( 0.001*pi*(1:M).*(1:M) ); plot( 1:M, y, ro- ) Use this vectorization idea to write 2 or 3 lines of code that will perform the same task as the following MATLAB script without using a for loop. (Note: there is a difference between the two operations xx*xx and xx.*xx when xx is a vector.) %--- make a plot of a weird signal N = 200; for :N xk(k) = k/50; rk(k) = sqrt( xk(k)*xk(k) ); sig(k) = exp(j*2*pi*rk(k)); end plot( xk, real(sig), mo- ) 4

5 3.4 Functions Functions are a special type of M-file that can accept inputs (matrices and vectors) and also return outputs. The keyword function must appear as the first word in the ASCII file that defines the function, and the first line of the M-file defines how the function will pass input and output arguments. The file extension must be lower case m as in my func.m. See Section B.5 in Appendix B for more discussion. The following function has a few mistakes. Before looking at the correct one below, try to find these mistakes (there are at least three): matlab mfile [xx,tt] = badcos(ff,dur) %BADCOS Function to generate a cosine wave % usage: % xx = badcos(ff,dur) % ff = desired frequency in Hz % dur = duration of the waveform in seconds % tt = 0:1/(100*ff):dur; %-- gives 100 samples per period badcos = cos(2*pi*freeq*tt); The corrected function should look something like: function [xx,tt] = goodcos(ff,dur) tt = 0:1/(100*ff):dur; %-- gives 100 samples per period xx = cos(2*pi*ff*tt); Notice the word function in the first line. Also, freeq has not been defined before being used. Finally, the function has xx as an output and hence xx should appear in the left-hand side of at least one assignment line within the function body. The function name is not used to hold values produced in the function. 4 Warmup: Complex Exponentials In the Pre-Lab part of this lab, you learned how to write M-files. In this section, you will write two functions that can generate sinusoids or sums of sinusoids. 4.1 M-file to Generate a Sinusoid Write a function that will generate a single sinusoid, x(t) = A cos(ωt + φ), by using four input arguments: amplitude (A), frequency (ω), phase (φ) and duration (dur). The function should return two outputs: the values of the sinusoidal signal (x) and corresponding times (t) at which the sinusoid values are known. Make sure that the function generates 20 values of the sinusoid per period. Call this function one cos(). Hint: use goodcos() from part (a) as a starting point. Demonstrate that your one cos() function works by plotting the output for the following parameters: A = 95, ω = 200π rad/sec, φ = π/5 radians, and dur = seconds. Be prepared to explain to the lab instructor features on the plot that indicate how the plot has the correct period and phase. What is the expected period in millisec? Instructor Verification (separate page) 4.2 Sinusoidal Synthesis with an M-file: Different Frequencies Since we will generate many functions that are a sum of sinusoids, it will be convenient to have a function for this operation. To be general, we will allow the frequency of each component (f k ) to be different. The following expressions are equivalent if we define the complex amplitudes X k as X k = A k e jφ k. { N } x(t) = Re X k e j2πf kt (7) 5

6 x(t) = N A k cos(2πf k t + φ k ) (8) Write the Function M-file Write an M-file called syn sin.m that will synthesize a waveform in the form of (7). Although for loops are rather inefficient in MATLAB, you must write the function with one loop in this lab. The first few statements of the M-file are the comment lines they should look like: function [xx,tt] = syn_sin(fk, Xk, fs, dur, tstart) %SYN_SIN Function to synthesize a sum of cosine waves % usage: % [xx,tt] = syn_sin(fk, Xk, fs, dur, tstart) % fk = vector of frequencies % (these could be negative or positive) % Xk = vector of complex amplitudes: Amp*eˆ(j*phase) % fs = the number of samples per second for the time axis % dur = total time duration of the signal % tstart = starting time (default is zero, if you make this input optional) % xx = vector of sinusoidal values % tt = vector of times, for the time axis % % Note: fk and Xk must be the same length. % Xk(1) corresponds to frequency fk(1), % Xk(2) corresponds to frequency fk(2), etc. The MATLAB syntax length(fk) returns the number of elements in the vector fk, so we do not need a separate input argument for the number of frequencies. On the other hand, the programmer (that s you) should provide error checking to make sure that the lengths of fk and Xk are the same. See help error. Finally, notice that the input fs defines the number of samples per second for the cosine generation; in other words, we are no longer constrained to using 20 samples per period. Include a copy of the MATLAB code with your lab report Default Inputs You can make the last input argument(s) take on default values if you use the nargin operator in MATLAB. For example, tstart can be made optional by including the following line of code: Testing if nargin<5, tstart=0, end %--default value is zero In order to use this M-file to synthesize harmonic waveforms, you must choose the entries in the frequency vector to be integer multiples of some desired fundamental frequency. Try the following test and plot the result. [xx0,tt0] = syn_sin([0,100,250],[10,14*exp(-j*pi/3),8*j],10000,0.1,0); %-Period =? Measure the period of xx0 by hand. Then compare the period of xx0 to the periods of the three sinusoids that make up xx0, and write an explanation on the verification sheet of why the period of xx0 is longer. Instructor Verification (separate page) 6

7 5 Lab Exercise: Representation of Sinusoids with Complex Exponentials In MATLAB consult help on exp, real and imag. Be aware that you can also use the DSP First function zprint to print the polar and rectangular forms of any vector of complex numbers. (a) Generate the signal x(t) = Re{2e jπt + 2e jπ(t 1.25) + (1 j)e jπt } and make a plot versus t. Use the syn sin function and take a range for t that will cover three periods starting at t = 1 2 secs. Include the MATLAB code with your report. (b) From the plot of x(t) versus t, measure the frequency, phase and amplitude of the sinusoidal signal by hand. Show annotations on the plots to indicate how these measurements were made and what the values are. Compare to the calculation in part (c). (c) Use the phasor addition theorem and MATLAB to determine the magnitude and phase of x(t). 6 Lab Exercise: Multipath Fading In a mobile radio system (e.g., cell phones or AM radio), there is one type of degradation that is a common problem. This is the case of multipath fading caused by reflections of the radio waves which interfere destructively at some locations. Consider the scenario diagrammed in Fig. 1 where a vehicle traveling on the roadway receives signals from two sources: directly from the transmitter and reflections from another object such as a large building. This multipath problem can be modeled easily with sinusoids. The total received signal at the vehicle is the sum of two signals which TRANSMITTER Velocity of Propagation = c (velocity of light) y (0,d t ) REFLECTED PATH DIRECT PATH REFLECTOR Strength = 100% Phase Shift = 180 o (d xr,d yr ) VEHICLE x v x Figure 1: Scenario for multipath in mobile radio. A vehicle traveling on the roadway (to the right) receives signals from two sources: the transmitter and a reflector located at (d xr, d yr ). are themselves delayed versions of the transmitted signal, s(t). (a) The amount of the delay (in seconds) can be computed for both propagation paths. First of all, consider the direct path. The time delay is the distance divided by the speed of light ( m/s). Write a mathematical expression for the time delay in terms of the vehicle position x v and the transmitter location (0, d t ). Call this delay time t 1 and express it as a function of x v, i.e., t 1 (x v ). (b) Now write a mathematical formula for the time delay of the signal that travels the reflected path from the transmitter at (0, d t ) to the reflector at (d xr, d yr ) and then to the vehicle at (x v, 0). Call this delay time t 2 and make sure that you also express it as a function of x v, i.e., t 2 (x v ). In this case, you must add together two delays: transmitter to reflector and then reflector to vehicle. 7

8 (c) The received signal at the vehicle, r v (t), is the sum of the two delayed copies of the transmitter signal r v (t) = s(t t 1 ) s(t t 2 ) where s( ) is the transmitted signal, and the minus sign accounts for fact that the reflection reverses the phase of the reflected signal. 3 Assume that the source signal s(t) is a zero-phase unit-amplitude sinusoid at frequency f 0 = 150 MHz; and also assume that the transmitter is located at (0, 1500) meters and the reflector at (100, 900) meters. Then the received signal is the sum of two sinusoids. Make a plot of r v (t) when the vehicle position is x v = 0 meters. Plot 3 periods of r v (t) and then measure its maximum amplitude. (d) Repeat the work in part (c), but use the complex amplitudes instead. Explain how a single complex addition, followed by a magnitude operation can be used to find the amplitude of r v (t). (e) Our aim in the rest of this lab is to make a plot of signal strength versus vehicle position (x v ). One approach would be to repeat the process in part (c) for every position, i.e., generate the resultant sinusoid and measure its amplitude. However, that would be a huge waste of computation. Instead, the complex amplitude approach in part (d) is a much more efficient method. Derive a mathematical expression for complex amplitudes of each delayed sinusoid as a function of position x v. Then write a MATLAB program that will generate the time delays, form the complex amplitudes, and then add the complex amplitudes. Have the MATLAB program loop through the entire set of vehicle positions specified in part (f) below. Include this MATLAB code in your report and explain how it works. Vectorization: It is likely that your previous programming skills would lead you to write a loop to do this implementation. The loop would run over all possible values of x v, and would add the two complex amplitudes calculated at each x v. However, there is a much more efficient way in MATLAB, if you think in terms of vectors (which are really lists of numbers). In the vector strategy, you would make a vector containing all the vehicle positions; then do the distance and time delay calculations to generate a vector of time delays; next, a vector of complex amplitudes would be formed. In each of these calculations, only one line of code is needed and no loops. You need two vectors of complex amplitudes (one for the direct path and the other for the reflected path), so the whole process is done twice and then you can perform a vector add of the two complex amplitudes. (f) Utilize the MATLAB program from the previous part to generate a plot of signal strength versus vehicle position, x v, over the interval from 0 meters to +300 meters. 4 Assume that signal strength is defined to be the peak value of the received sinusoid, r v (t). State how you get the peak value from the complex amplitude. (g) Explain your results from the previous part. In particular, what are the largest and smallest values of received signal strength? Why do we get those values? Are there vehicle positions where we get complete signal cancellation? If so, determine those vehicle positions. 3 For simplicity we are ignoring propagation losses: When a radio signal propagates over a distance R, its amplitude will be reduced by an amount that is inversely proportional to R 2. 4 MATLAB works on vectors so you will have to produce a vector of positions starting at 0 and ending at +300 with a spacing that is small enough to capture all the variations in signal strength. 8

9 Lab 02a INSTRUCTOR VERIFICATION SHEET For each verification, be prepared to explain your answer and respond to other related questions that the lab TA s or professors might ask. Turn this page in at the end of your lab period. Name: Date of Lab: Part 4.1 Demonstrate that your one cos function is correct by plotting a sinusoidal signal with the given parameters. Use the space below to calculate the period of the sinusoid. Verified: Date/Time: Part Show that your syn sin.m function is correct by running the test in Section and plotting the result. Measure the period(s) and explain why the period of xx0 is longer that the periods of the signals used to form xx0. Write your explanations in the space below. Verified: Date/Time: 9

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